Step 1

Adjust oven rack to upper-middle position and heat oven to 450 degrees. Combine beef, mushrooms, onion, garlic, thyme, 1½ teaspoons salt, and 1½ teaspoons pepper in Dutch oven. Cook over medium-high heat until nearly all liquid has evaporated, 25 to 28 minutes, stirring occasionally and breaking up meat with spoon.

Step 2

Stir in flour until fully incorporated and cook for 1 minute. Stir in milk and broth and bring to simmer, scraping up any browned bits. Cook until mixture is slightly thickened, about 3 minutes. Off heat, stir in Parmesan. Transfer mixture to 13 by 9-inch baking dish.

Step 3

Sprinkle peas and corn evenly over beef mixture. Lightly arrange tater tots in even layer over top, but do not press into mixture (you may have extra tater tots). Bake until tater tots are deep golden brown and filling is bubbling, 35 to 38 minutes, rotating dish halfway through baking. Let cool for 15 minutes before serving.

Step 4

TO MAKE AHEADAt end of step 2, let beef mixture cool completely, then cover with aluminum foil and refrigerate for up to 24 hours. To serve, bake beef mixture, covered, until hot in center, 15 to 20 minutes. Remove foil, stir beef mixture, sprinkle with peas and corn, arrange tater tots on top, and bake as directed in step 3.